On Complex Kinematics and Relativity

• Published in: Advances in applied Clifford algebras 2022, Vol.32 (3), Article 38
• Main Author: Brezov, Danail
• Format: Article
• Language: English
• Subjects: 2020; Angular velocity; Applications of Mathematics; August 3-7; Body kinematics; Electromagnetic fields; Electromagnetism; Group theory; Hefei; Kinematics; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Relativity; Rigid structures; T.C. ICCA 12; Theoretical; Theory of relativity
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-022-01220-4

Using the well known complex representation of the proper Lorentz group SO + ( 3 , 1 ) ≅ PSL ( 2 , C ) ≅ SO ( 3 , C ) we study some Coriolis type effects in Special Relativity and Electromagnetism in close analogy with the more traditional kinematical treatment of the group of spatial rotations. Namely, we begin with the Clifford group of R 3 viewed as a complexification of H × and consider the associated Maurer-Cartan form which yields a complex-valued analogue of the angular velocity characterizing the action of SO ( 3 ) in rigid body kinematics. It appears in a linear ODE’s written in biquaternion form or a Ricatti equation if one works with its projective version instead, providing a far richer structure compared to the real case without imposing serious technical obstruction at least in the decomposable setting, which is our main emphasis due to its importance in physics. There are several distinct terms in the non-commutative part of the so obtained connection describing well known effects named after Coriolis, Thomas, Hall and Sagnac. We also consider a restriction to the so-called Wigner little groups SO ( 3 ) , SO + ( 2 , 1 ) and E ( 2 ) discussing algebraic properties of the electromagnetic field. Some familiar constructions such as geometric phases, Hopf bundles and the Fubini-Study form appear naturally with this approach.